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2 votes
1 answer
91 views

Are the injections of a coproduct a cover in the canonical pretopology?

Assume we're in a category $C$ with all pullbacks and finite coproducts. Recall that the canonical coverage of $C$ is the finest Grothendieck (pre) topology for which all representables are sheaves. A …
Joey Eremondi's user avatar
6 votes
1 answer
368 views

Relationship between canonical topology on a topos and its site of definition

The canonical (Grothendieck) topology for a category $C$ is the largest (finest) topology such that every representable presheaf over $C$ is a sheaf. According to First Order Categorical Logic Lemma 1 …
Joey Eremondi's user avatar
2 votes
1 answer
148 views

Is the slice of a subcanonical site also subcanonical?

A subcanonical site is one for which every representable functor is a sheaf. For a subcanonical site $C$, the fundamental theorem of topos theory says that there is an equivalence $Sh(C/c)\cong Sh(C)/ …
Joey Eremondi's user avatar
3 votes
0 answers
208 views

How to read the definition of Grothendieck Pretopology in SGA4?

In SGA4, the first axiom of a Grothendieck pretopology is given as: PT0: Pour tout objet $X$ de $C$, les morphismes des familles de morphismes de $Cov(𝑋)$ sont quarrables. (Rappelons qu’un morphisme …
Joey Eremondi's user avatar